A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function (Q879065)

For an integer \(q>0\) and arbitrary integer \(h\), let \[ C(h,q)= \sum_{\substack{ 1 \leq a \leq q\\ (a,q)=1}} \bigg(\bigg(\frac{{\overline a}}{q}\bigg)\bigg)\bigg(\bigg(\frac{ah}{q}\bigg)\bigg), \] and \[ K(m,n;q)= \sum_{\substack{ b=1\\ (b,q)=1}} e^{2 \pi i (mb+n{\overline b})/q}, \] where \(((x))=x-[x]-1/2\) if \(x\) is not an integer, and \(((x))=0\) if \(x\) is an integer, and \({\overline a}\) is defined by the equation \(a {\overline a}=1 \mod q\), denote the Cochrane and Kloosterman sums, respectively. The author studies the asymptotic properties of the hybrid mean value of the Kloosterman sums with the weight of Hurwitz zeta-function \(\zeta(s;a)\), \(s=\sigma+it\), \(0<a\leq 1\), and the Cochrane sums. For any positive integer \(q\), \(1 \leq h\leq q\), \((h,q)=1\), the asymptotic formula \[ \sum_{\substack{ 1 \leq a \leq q\\ (a,q)=1}}\;\sum_{\substack{ 1 \leq b \leq q\\ (b,q)=1}} K(a,1;q)\zeta\bigg(\frac{1}{2},\frac{b}{q}\bigg)C(ab,q) =-\frac{q^{3/2}\varphi(q)}{2 \pi^2}\prod_{p \| q}\bigg(1-\frac{1}{p(p-1)}\bigg)+O(q^{2+\varepsilon}), \] where \(\varepsilon\) denotes any fixed positive number, \(\prod_{p\| q}\) denotes the product over all prime divisors \(p\) of \(q\) such that \(p| q\) and \(p^2 \nmid q\), and \(\varphi(q)\) is the Euler function, is proved.